Trace of Linear Transformations

Question

I need to find the trace of the following powers of linear transformation which is already given.

My doubt is that Is there any better way than finding the matrix of linear transformation and then taking it's powers. Because it will involve quite a lot of calculation as we will have a 10× 10 matrix. Am I missing any trick?? Thanks and regards.

Answer 1

You have a 10x10 matrix indeed, but a really specific one.

Hint: Note that

• $T(a_0)=a_0$
• $X$ and $X^2$ coefficients are "mixed up"
• So are $X^3$, $X^4$ and $X^5$ coefficients
• Same with $X^i, i\in \{6,\ldots,9\}$

Treat each "block" separatly, and arrange them to form a block diagonal matrix